> was bad)  With this in mind, I guess
> in the remark this would correspond to something like considering the
> "enriched"
> presheaves with some chosen model structure.  But I'm getting a bit out of
> my depth here
> so maybe someone can chime in to help me out ...

The model structure that you are after here, for excisive functors,
modeled on the pointed topologically enriched category of finite
pointed CW-complex is discussed in

Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley,
"Model categories of diagram spectra"
https://ncatlab.org/nlab/show/Model+categories+of+diagram+spectra

This is the topologically enriched version of Lydakis' simplicially
enriched model structure for excisive functors

https://ncatlab.org/nlab/show/model+structure+for+excisive+functors